Strict Mathematical Theory of Equilibrium Deviation Field for Navier-Stokes Equations

Based on the steady smooth regular solutions of Navier-Stokes equations, this paper constructs a complete theoretical system of equilibrium deviation field. Through flow field decomposition, the original governing equations are transformed into closed evolution equations dominated by deviation field variables. The full flow domain is divided into near-equilibrium region and effective regulation region by threshold constraint,and complete spatiotemporal boundary conditions are formulated. With existing partial differential equation regularity theories, the uniform boundedness of deviation field and its gradient terms is strictly proved. A piecewise dynamic regulation coefficient is established, which is globally well-defined without singularity and satisfies uniform boundedness condition. This regulation strategy keeps strict equivalence with original equations,and possesses local adaptive optimality derived from gradient-driven mechanism, which can effectively restrain nonlinear flow behaviors. The whole theoretical framework is logically self-consistent and mathematically rigorous. All discussions are restricted within the proven smooth existence domain of Navier-Stokes equations,without involving the demonstration of global regularity. This work provides a novel standardized analysis paradigm for flow evolution research and nonlinear dynamic regulation.